Finite Group Databasebeta

← #27150020

Subgroups — #27150020

Order 12416.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D97C211yesmaximal, characteristic
2C2×#15690786C212yesmaximal, complemented
3#20260607C212yesmaximal, complemented
4#20260606C214yesmaximal, complemented
5#20260595C211yesmaximal, characteristic, complemented
6#20260592C212yesmaximal, complemented
7#20260590C212yesmaximal
8C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C23×Dic97C2211yescharacteristic
11C22×C4×D97—28no—
12C22×C4×D97—24no—
13C22×C4×D97C2212yes—
14C22×C4×D97C2212yes—
15#15690786—28no—
16C2×C4⋊Dic97C2212yescomplemented
17C2×#13095490C2214yescomplemented
18C2×C22⋊Dic97C2212yes—
19C2×#13095488C2214yes—
20C2×#13095488C2214yes—
21#15690717C2214yescomplemented
22#15690717C2212yescomplemented
23C22×Q8×C97C2211yescharacteristic, complemented
24C23×C4×C97C2211yescharacteristic
25C22⋊Q8×C97—24no—
26C2×C4⋊C4×C97C2211yescharacteristic, complemented
27C2×C4⋊C4×C97C2212yescomplemented
28C2×C22⋊C4×C97C2212yes—
29C23×D97—22no—
30C23×D97—22no—
31C23×D97—24no—
32C23×D97—24no—
33C23×D97Q814yescomplemented
34C23×D97C2312yes—
35C22×Dic97—22no—
36C22×Dic97—24no—
37C22×Dic97C2314yes—
38C22×Dic97C2312yes—
39C2×C4×D97—232no—
40C2×C4×D97—24no—
41C2×C4×D97—24no—
42C2×C4×D97—24no—
43C2×C4×D97—24no—
44C2×C4×D97—24no—
45C2×C4×D97D418yes—
46C4⋊Dic97—24no—
47#13095490—28no—
48C22⋊Dic97—24no—
49#13095488—28no—
50#13095488—28no—
51C24×C97C2311yescharacteristic
52C2×Q8×C97—24no—
53C2×Q8×C97—22no—
54C22×C4×C97—24no—
55C22×C4×C97D414yes—
56C22×C4×C97C2312yes—
57C22×C4×C97C2312yes—
58C22×C4×C97C2311yescharacteristic
59C22×C4×C97C2311yescharacteristic
60C4⋊C4×C97—22no—
61C4⋊C4×C97—24no—
62C22⋊C4×C97—24no—
63C23×C97—22no—
64C23×C97—22no—
65C23×C97C2×D412yes—
66C23×C97C2×Q812yes—
67C23×C97#4112yes—
68C23×C97C2411yescharacteristic
69C22×D97—28no—
70C22×D97—216no—
71C22×D97—28no—
72C22×D97—28no—
73C22×D97—22no—
74C22×D97—22no—
75C22×D97—22no—
76C22×D97—22no—
77C22×D97—216no—
78C22×D97C2×D414yes—
79C22×D97C2×Q814yes—
80C22×D97#4114yes—
81Q8×C97—44no—
82C2×C4×C97—22no—
83C2×C4×C97—28no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—24no—
87C2×C4×C97—24no—
88C2×C4×C97C2×D414yes—
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2×D412yes—
91C2×Dic97—28no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—24no—
95C2×Dic97—24no—
96C2×Dic97—24no—
97C2×Dic97C2×D414yes—
98C4×D97—216no—
99C4×D97—216no—
100C22×C97—22no—
101C22×C97—24no—
102C22×C97—22no—
103C22×C97—24no—
104C22×C97C22⋊Q814yescomplemented
105C22×C97C22×D412yes—
106C22×C97C22×D411yescharacteristic
107C22×C97C22×Q811yescharacteristic
108C22×C97C2×#4112yes—
109C22×C97C2×#4111yescharacteristic
110C2×D97—232no—
111C2×D97—24no—
112C2×D97—24no—
113C2×D97—24no—
114C2×D97—24no—
115C2×D97—24no—
116C2×D97—24no—
117C2×D97C22⋊Q818yescomplemented
118C4×C97—22no—
119C4×C97—42no—
120C4×C97—42no—
121C4×C97#12014yes—
122Dic97—24no—
123Dic97—44no—
124C2×C97—24no—
125C2×C97C2×#12011yescharacteristic
126C2×C97C2×C22⋊Q812yes—
127C2×C97C2×C22⋊Q811yescharacteristic
128C2×C97#52511yescharacteristic
129C2×C97#52512yes—
130D97—28no—
131#2848—971nomaximal, Sylow
132C97#284811yescharacteristic, complemented, Sylow
133C24×C4—971no—
134C2×C22⋊Q8—972no—
135C2×C22⋊Q8D9711yescharacteristic, complemented
136#385—974no—
137#385—972no—
138#380—971no—
139#380—972no—
140#379—972no—
141C25—971no—
142C22×Q8C2×D9711yescharacteristic, complemented
143C23×C4—972no—
144C23×C4—971no—
145C23×C4—972no—
146C23×C4—1948no—
147C23×C4—1944no—
148C23×C4C2×D9711yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—1948no—
151C2×C4⋊C4—974no—
152C2×C4⋊C4—972no—
153C2×C4⋊C4C2×D9711yescharacteristic, complemented
154C2×C4⋊C4C2×D9712yescomplemented
155C2×C22⋊C4—972no—
156C2×C22⋊C4—974no—
157C2×C22⋊C4—974no—
158C2×C22⋊C4C2×D9712yes—
159#95—974no—
160#95—972no—
161C24—972no—
162C24—974no—
163C24—1944no—
164C24—1944no—
165C24—1942no—
166C24—1942no—
167C24C22×D9711yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—972no—
172C22×C4—978no—
173C22×C4—974no—
174C22×C4—1944no—
175C22×C4—1944no—
176C22×C4—19432no—
177C22×C4—1944no—
178C22×C4—1942no—
179C22×C4—1944no—
180C22×C4—1944no—
181C22×C4—1944no—
182C22×C4C22⋊D9714yes—
183C22×C4C22×D9712yes—
184C22×C4C22×D9712yes—
185C22×C4C22×D9711yescharacteristic
186C22×C4C22×D9711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—1948no—
190C4⋊C4—1944no—
191C22⋊C4—24no—
192C22⋊C4—1948no—
193C22⋊C4—1948no—
194C22⋊C4—1944no—
195C23—22no—
196C23—22no—
197C23—974no—
198C23—974no—
199C23—974no—
200C23—19416no—
201C23—1948no—
202C23—1948no—
203C23—19416no—
204C23—1948no—
205C23—1942no—
206C23—1942no—
207C23—1942no—
208C23—1942no—
209C23C2×C22⋊D9712yes—
210C23Q8×D9712yes—
211C23#1309551212yes—
212C23C23×D9711yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—974no—
221C2×C4—19416no—
222C2×C4—1944no—
223C2×C4—1944no—
224C2×C4—1944no—
225C2×C4—1948no—
226C2×C4—1942no—
227C2×C4—1942no—
228C2×C4—19416no—
229C2×C4D4×D9714yes—
230C2×C4C2×C22⋊D9712yes—
231C2×C4C2×C22⋊D9712yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—978no—
237C22—19432no—
238C22—1944no—
239C22—1944no—
240C22—1944no—
241C22—1944no—
242C22—1944no—
243C22—1944no—
244C22#1569078614yescomplemented
245C22C2×D4×D9712yes—
246C22C22×C22⋊D9711yescharacteristic
247C22C2×Q8×D9711yescharacteristic
248C22#41×D9712yes—
249C22C2×#1309551211yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—3884no—
254C4—1944no—
255C4#1569077814yes—
256C2—24no—
257C2—1948no—
258C2C2×#1569077811yescharacteristic
259C2C22⋊Q8×D9712yes—
260C2C2×#1569078611yescharacteristic
261C2#2026124211yescharacteristic
262C2#2026124812yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×C22⋊Q8×C97C2
prime radicalC24×D97C22
hypercenterC2×C22⋊Q8D97
hypocenterC97#2848
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#15690786C2semidirect
3. #20260607C2semidirect
4. #20260606C2semidirect
5. #20260595C2semidirect
6. #20260592C2semidirect
8. C2×C22⋊Q8×C97C2semidirect
16. C2×C4⋊Dic97C22semidirect
17. C2×#13095490C22semidirect
21. #15690717C22semidirect
22. #15690717C22semidirect
23. C22×Q8×C97C22semidirect
26. C2×C4⋊C4×C97C22semidirect
27. C2×C4⋊C4×C97C22semidirect
33. C23×D97Q8semidirect
104. C22×C97C22⋊Q8semidirect
117. C2×D97C22⋊Q8semidirect
132. C97#2848semidirect
135. C2×C22⋊Q8D97semidirect
142. C22×Q8C2×D97semidirect
153. C2×C4⋊C4C2×D97semidirect
154. C2×C4⋊C4C2×D97semidirect
244. C22#15690786semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C4×D972
C2×#156907862
#202606072
#202606062
#202605952
#202605922
#202605902
C2×C22⋊Q8×C972
#284897